Journal ArticleDOI
Nonlinear convection in a rotating layer: Amplitude expansions and normal forms
John Guckenheimer,Edgar Knobloch +1 more
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In this article, a two-dimensional convection in a horizontal layer of Boussinesq fluid rotating about a vertical axis is studied, where the dispersion relation describing the stability of the conductive solution has two zero eigenvalues.Abstract:
Two-dimensional convection in a horizontal layer of Boussinesq fluid rotating about a vertical axis is studied. For certain choices of the parameters the dispersion relation describing the stability of the conductive solution has two zero eigenvalues. For nearby parameter values nonlinear solutions are accessible analytically, using either the method of normal forms or an amplitude expansion. The results provide a complete description of the transitions between oscillatory and steady convection as functions of the Rayleigh and Taylor numbers near their critical values.read more
Citations
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A new approach to energy theory in the stability of fluid motion
Journal ArticleDOI
Convection in a rotating cylinder. Part 1. Linear theory for moderate Prandtl numbers
TL;DR: In this paper, the onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied, where the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns.
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Chaos in models of double convection
TL;DR: In this article, a unified approach to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour is presented.
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Transitions to chaos in two-dimensional double-diffusive convection
TL;DR: In this paper, the authors studied the transition from nonlinear periodic oscillations to temporal chaos and revealed sequences of period-doubling bifurcations, leading to aperiodicity, as the thermal Rayleigh number R(T) is increased.
Journal ArticleDOI
A Nonlinear Analysis of the Stabilizing Effect of Rotation in the Benard Problem
TL;DR: In this paper, the authors used a generalized energy to study the stabilizing effect of rotation in the Benard problem and found that the nonlinear stability boundary is in very close agreement with the experiments of Rossby (Rossby, H. T. and A. J. Jansen, 1969) who predicted sub-critical instabilities for high Taylor numbers for fluids with Prandtl number greater than or equal to 1.
References
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Book
The Hopf Bifurcation and Its Applications
TL;DR: The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value as mentioned in this paper.
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Cellular convection with finite amplitude in a rotating fluid
TL;DR: In this article, it is shown that the boundary of a steady convection cell is distorted by the rotation in such a way that the wave length of the cell measured along the distorted boundary is equal to the wavelength of the non-rotating cell.
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Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields
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Nonlinear periodic convection in double-diffusive systems
TL;DR: In this article, the authors studied two examples of two-dimensional nonlinear double-diffusive convection (thermohaline convection and convection in an imposed vertical magnetic field) in the limit where the onset of marginal overstability just precedes the exchange of stabilities.
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The Instability of a Layer of Fluid Heated below and Subject to Coriolis Forces. II
TL;DR: In this paper, the authors examined the stability of a horizontal layer of fluid heated below, subject to an effective gravity acting (approximately) in the direction of the vertical and the Coriolis force resulting from a rotation of angular velocity Ω about a direction making an angle ϑ with the vertical.